3.8 Related rates 189man is rising -* units per second and is therefore falling -- units per
second, and our problem is solved. It can be insisted that our solution
of the problem would have been more easily understood if we had used
the more elaborate symbols x(t) or fl(t) and y(t) or f2(t) instead of x and
y to denote distances. Thus we could have written[x(t)]2 + [y(t)]2 = 100, x(t)x'(t) + Y(t)Y'(t) = 0
or
[fl(t)]2 + [.12(t)]2 =100, fl(t)f'(t) +f2(t)f2(t) = 0
instead of (3.83) and (3.84). One who wishes to do so may insist that
(3.83) and (3.84) abbreviate more meaningful formulas just as the sym-
bols AA and AAA abbreviate Alcoholics Anonymous and American
Automobile Association. It is, however, required that we learn the
abbreviations to expedite our work and to enable us to understand others
who use the abbreviations.
When we are interested in problems involving rates of change of the
volume V and the radius r of a sphere, we start with the formula(3.85) V = 4-7rr3 or P(t) _ 4-ir[r(t)]3.Supposing that V and r are differentiable functions of time t, we can
differentiate to obtain(3.851) dy
t= 4ar2dr
tWhen numerical values are assigned to two of the three quantities r,
dr/dt, dV/dt, we can solve (3.851) for the remaining quantity.
We need very little information about the external world to appreciate
the idea that if an appropriate piston is pushed into a closed cylinder con-
taining a gas, then the volume Y of the confined gas will decrease and
the pressure p exerted by the confined gas will increase. In appropriate
circumstances, calculations can be based upon the formula
(3.86) pY = c,
where p and V are differentiable functions of t and c is a constant. Dif-
ferentiation with respect to t gives the formula
(3.861) pdY-E-Ydp= O,which involves four numbers. When three of these numbers are known,
we can calculate the remaining one.
If a particle is moving along the graph of the equation y = x2 in such
a way that its coordinates x, y are differentiable functions of t, then
(3.87) dy= 2dx
dt x dt